I am asked to write explicitly, in spherical coordinates, a boundary problem that models the temperature $u$ wih regard to the sun, considered nearly spherical, with radius R and surface temperatureT = 5777 K. In our model, we disregard convection and other complicating effects, and assume the temperature u in the interior of the star is stationary (no timedependence), a radial function (u = u(r)), and governed by the Poisson equation $\Delta u = f$. Here, the source term f is a $C^∞$ function that models the heat production in the Sun due to nuclear fusion.
My model is the following:
$\Delta u= \frac{d^2}{dr^2}u+\frac{2}{r} \frac{d}{dr}u = f$
$u(R)=T=5777 K$
where $u(R)=T=5777 K$ is the Dirichlet boundary condition
The heat distribution of the Sun can be modelled by the Poisson equation:
$$ \Delta u = f$$
To write in spherical coordinates a boundary problem that models the temperature $u$ we can use the spherical Laplacian coordinates for the problem:
$$ \Delta u= \frac{d^2}{dr^2}u+\frac{2}{r} \frac{d}{dr}u = f$$
Since $u=u(r)$ is only a function of the radius the Beltrami operator can be neglected.
Is it correct in your opinion?